Inverse Element is Power of Order Less 1

Theorem

Let $G$ be a group whose identity is $e$.

Let $g \in G$ be of finite order.

Then:

$\order g = n \implies g^{n - 1} = g^{-1}$

$\ds \order g$

$=$

$\ds n$

$\ds \leadsto \ \ $

$\ds g^n$

$=$

$\ds e$

$\ds \leadsto \ \ $

$\ds g^n g^{-1}$

$=$

$\ds e g^{-1}$

$\ds \leadsto \ \ $

$\ds g^{n - 1}$

$=$

$\ds g^{-1}$

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups: Example $101$
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Definition $3.9$: Remark $2$